You are always better off cleaning up your data prior to training. There was quite a bit of literature related to how errors in the training data affect the quality of a classifier. This paper was somewhat later than others but was very well received.
Brodley, Friedl. "Identifying Mislabeled Training Data" 1999. J. Art. Intel. Research. 132-167.
They conclude (consistent with some of the earlier papers) that a one percent error in the training data can affect the quality of the classifier. Note that this paper and others only dealt with the situation where the data where fairly clean but for which some of the records in the training data were misclassified. Other literature deals with some of the general methods for cleaning up the data.
If the noise is related with wrongly labels it will affect precision of classification, but if you have thousand of images and only a percentage of mislabeling lower than 1% the effect in precision will be low.
If the noise is in each image, something like gaussian noise or occlusions it may turn the classifier "strong" if the noise is small.
If you can control the amount of noise on the images, you should try something like Denoising Autoencoders to improve their quality.